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Wednesday, January 3, 2024

Heat conductance of the quantum Hall bulk - Nature.com

Abstract

The quantum Hall effect is a prototypical realization of a topological state of matter. It emerges from a subtle interplay between topology, interactions and disorder1,2,3,4,5,6,7,8,9. The disorder enables the formation of localized states in the bulk that stabilize the quantum Hall states with respect to the magnetic field and carrier density3. Still, the details of the localized states and their contribution to transport remain beyond the reach of most experimental techniques10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31. Here we describe an extensive study of the bulk’s heat conductance. Using a novel ‘multiterminal’ short device (on a scale of 10 µm), we separate the longitudinal thermal conductance, \({\kappa }_{xx}T\) (owing to the bulk’s contribution), from the topological transverse value \({\kappa }_{xy}T\) by eliminating the contribution of the edge modes24. When the magnetic field is tuned away from the conductance plateau centre, the localized states in the bulk conduct heat efficiently (\({\kappa }_{xx}T\propto T\)), whereas the bulk remains electrically insulating. Fractional states in the first excited Landau level, such as the \(\nu =7/3\) and \(\nu =5/2\), conduct heat throughout the plateau with a finite \({\kappa }_{xx}T\). We propose a theoretical model that identifies the localized states as the cause of the finite heat conductance, agreeing qualitatively with our experimental findings.

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Fig. 1: Device and longitudinal thermal conductance at filling ν = 2.
Fig. 2: Temperature dependence of the longitudinal thermal conductance coefficient κxx at ν = 2.
Fig. 3: Illustration of the model used to calculate the bulk’s thermal conductance.
Fig. 4: Thermal conductance of the bulk at fillings ν = 7/3 and ν = 5/2.

Data availability

All relevant data have been provided in this paper. Additional information related to this work is available from the corresponding author upon reasonable request. Source data are provided with this paper.

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Acknowledgements

We thank A. D. Mirlin for fruitful discussions. A.G. and E.B. acknowledge support from the Israel Science Foundation (Quantum Science and Technology Grant 2074/19) and the Deutsche Forschungsgemeinschaft (Grant CRC 183). M.H. acknowledges support from the European Research Council (the European Union’s Horizon 2020 Research and Innovation Program Grant 833078). A.S. acknowledges support from the Israel Science Foundation (Quantum Science and Technology Grant 2074/19), the Deutsche Forschungsgemeinschaft (Grant CRC 183 and Project C02) and the European Research Council (the European Union’s Horizon 2020 Research and Innovation Program Grants 788715 and 817799 and Project LEGOTOP).

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Authors and Affiliations

  1. Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel

    Ron Aharon Melcer, Arup Kumar Paul, Priya Tiwari, Vladimir Umansky & Moty Heiblum

  2. Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel

    Ron Aharon Melcer, Avigail Gil, Arup Kumar Paul, Priya Tiwari, Vladimir Umansky, Moty Heiblum, Yuval Oreg, Ady Stern & Erez Berg

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  1. Ron Aharon Melcer
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Contributions

R.A.M. designed the experiment, fabricated the devices, performed the measurements and analysed the data. R.A.M., A.K.P. and P.T. performed length dependence measurements. M.H. supervised the experiment’s design, execution and data analysis. A.G., Y.O., A.S. and E.B. developed the theoretical model. V.U. grew the GaAs heterostructures. All authors contributed to the writing of the manuscript.

Corresponding author

Correspondence to Moty Heiblum.

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The authors declare no competing interests.

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Nature thanks Francois Parmentier, Bo Yang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended Data Fig. 1 Methodology of power measurement.

Measurement and analysis steps required to measure the heat flow and extract \({\kappa }_}\). As an example, we present \(\nu =2\) data, measured at \(B=6.1{\rm{T}}\), and base temperature of \({T}_{0}=15{\rm{mK}}\). (a) Raw noise data. The excess noise measured at \({A}_}\) and \({A}_}\) as a function of the sourced current \(I\) sourced from \({S}_{1}\) (while current \(-I\) is simultaneously sourced from \({S}_{2}\)). (b) Power-metre’s temperature as a function of the source’s temperature extracted from (a) using Eq. M7. The heating of the source from \({T}_{0}=15{\rm{mk}}\) to a temperature \({T}_} \sim 40{\rm{mK}}\) causes the slight increase of the PM’s temperature \({T}_} \sim 17{\rm{mK}}\), due to the finite \({\kappa }_}\). (c) Power-metre calibration; raw data. Noise measured at \({A}_}\) as a function of the direct heating of the PM, by current \({I}_}\) sourced from \({S}_{1}^}\) (while current \(-{I}_}\) is simultaneously sourced from \({S}_{2}^}\)). (d) Dissipated power (derived from Eq. M9) as a function of \({T}_}\). (e) By combining the main measurement (b) with the calibration (d), we can plot the power arriving to the PM as a function of the source temperature, and produce the plot presented in the main text Fig. 2a. A linear fit to the power vs. \({T}_}^{2}\) gives \({\kappa }_}\).

Extended Data Fig. 2 κxx for different device distance.

The dissipated power in the PM as a function of the source’s temperature squared, for two different PM-to-source-distances (measured in different devices), \(10\,\mu m\) and 20 \(\mu m.\) For both fractional states \(\nu =5/2\) (a) and \(\nu =7/3\) (b). We observe a decrease of the transferred heat with S-PM distance. This corresponds to \({\kappa }_}\) reducing from \({\kappa }_}=0.37\pm 0.03{\kappa }_{0}\) (\({\kappa }_}=0.24\pm 0.01{\kappa }_{0}\)) at \(10\,\mu m\) to \({\kappa }_}=0.17\pm 0.02{\kappa }_{0}\) (\({\kappa }_}=0.15\pm 0.01{\kappa }_{0}\)) at \(20\,\mu m\) for \(\nu =\frac{5}{2}\)(\(\nu =7/3\)). Data measured at the plateau centre at \({T}_{0}=10{\rm{mK}}\).

Extended Data Fig. 3 ‘Two terminal’ thermal conductance on the ν = 2 plateau, at T0 = 15mK.

(a) Power dissipated at the source, \({P}_}\), as a function of the source’s temperature squared, for different magnetic fields on the \(\nu =2\) plateau. The low temperature data (up to \(27{\rm{mK}}\)) is linearly fitted to extract the two-terminal thermal conductance, \({\kappa }_{2{\rm{T}}}\), which changes mildly with magnetic field. (b) Top-panel -\({\kappa }_{2{\rm{T}}}\), extracted from (a) as a function of the magnetic field (includes 2k0 due to donors), with an increase away from plateau centre due to the short bulk. Bottom-panel - \({\kappa }_}\), and \({G}_}\) as a function of the magnetic field (identical to Fig. 2b). It appears that the appearance of finite heat conductance through the bulk causes \({\kappa }_{2{\rm{T}}}\) to increase slightly.

Extended Data Fig. 4 Thermal conductance through the bulk of other QHE states.

Longitudinal electrical conductance (green with scale to the left) and longitudinal thermal conductance (red markers with scale on the right) plotted as a function of magnetic field on the plateaus of (a) \(\nu =3\), (b) \(\nu =4/3\) and (c) \(\nu =2\). The circular markers corresponds to the fitting results of \(P\) vs. \({T}_}^{2}\) (raw data presented in Extended Data Fig. 5), and the triangular markers correspond to \({\kappa }_}\) measured for a single source temperature of \({T}_}=50{\rm{mK}}\), and extracted according to Eq. M11.

Extended Data Fig. 5 Raw data used to extract κxx.

The coloured markers represent the power arriving to the PM as a function of the source temperature squared. The data is linearly fitted (coloured straight lines) to extract \({\kappa }_}\) (according to Eq. 1 of the main text). Showing the measured data for the results appearing in the main text and the supplementary information: (a) \(\nu =2\), (b) \(\nu =7/3\), (c), \(\nu =5/2\) and \(\nu =8/3\), (d) \(\nu =3\) and (e) \(\nu =4/3\).

Extended Data Fig. 6 Amplifier calibration.

Equilibrium noise as a function of the cryostat temperature (markers). The noise is linear in temperature, in agreement with the Johnson-Nyquist formula (Eq. M13). This allows us to calibrate the gain according to Eq. M14 (straight lines).

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Melcer, R.A., Gil, A., Paul, A.K. et al. Heat conductance of the quantum Hall bulk. Nature (2024). https://ift.tt/ohTx5LQ

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